Video Transcript
The graph of 𝑔 double prime is shown below. If 𝑔 prime of two equals zero, at what other value of 𝑥 on the interval from zero to 12 does 𝑔 prime of 𝑥 equal zero?
It’s helpful to write down what has been given and what is required to be found. We’re given 𝑔 double prime. That’s the second derivative of the function 𝑔 in graph form. We see that this graph is piecewise linear. And we also see some points that the graph goes through. And so we think that we could probably come up with an algebraic rule for 𝑔 double prime if we needed to.
We’re also given a single value of the first derivative of 𝑔. That’s 𝑔 prime. We’re told that 𝑔 prime of two is zero. And with this information, we are required to find at what other value of 𝑥 on the interval from zero to 12 𝑔 prime of 𝑥 equals zero. In other words, we need to solve 𝑔 prime of 𝑥 equals zero, finding the zeros of the first derivative of 𝑔, 𝑔 prime.
Now how might we do that? Well, a good intermediate step would be to find an expression for 𝑔 prime of 𝑥. Having found an expression for 𝑔 prime of 𝑥, hopefully we’ll be able to solve this equation. So how are we going to find an expression for 𝑔 prime of 𝑥 given one value of 𝑔 prime and also the graph of 𝑔 double prime of 𝑥?
The key thing to note is that 𝑔 double prime of 𝑥, the second derivative of 𝑔 of 𝑥, is also the first derivative of 𝑔 prime of 𝑥. All our information is now in terms of 𝑔 prime. We’re given the first derivative of this function and one value of the function. And we want to find the function itself. And we know how to do this by the fundamental theorem of calculus.
We can find the function 𝑓 of 𝑥, given the value of the function at 𝑎, 𝑓 of 𝑎, by adding the integral between 𝑎 and 𝑥 of its derivative 𝑓 prime of 𝑥 with respect to 𝑥. We’re interested in finding 𝑔 prime of 𝑥. And so we substitute 𝑔 prime of 𝑓. And we’re given the value of 𝑔 prime of two, so we substitute two for 𝑎. And so what we need to add is the integral between two and 𝑥 of the first derivative of 𝑔 prime, which is the second derivative of 𝑔, 𝑔 double prime of 𝑥 with respect to 𝑥.
So we have an expression for 𝑔 prime of 𝑥. And, in fact, we can simplify this as we know that 𝑔 prime of two is zero. So 𝑔 prime of 𝑥 is just equal to the integral between two and 𝑥 of 𝑔 double prime of 𝑥 with respect to 𝑥. And we substitute this into the equation that we are required to solve. And so the equation that we have to solve is the integral between two and 𝑥 of 𝑔 double prime of 𝑥 with respect to 𝑥 equals zero.
For what values of the upper limit 𝑥 of the integral does this equation hold? Well, we have a graph of the integrand 𝑔 double prime. So let’s look at that. We know that the lower limit of the integral is two. What should the upper limit be to make the integral zero?
Well, remember that the definite integral of a function is the assigned area between the graph of that function and the 𝑥-axis over a certain interval. To make the integral zero, the area above the 𝑥-axis must exactly cancel out the area below it. Can you see then what we ought to take the upper limit to be?
If we picked a value less than two, then all the area would be below the 𝑥-axis. There’s no way that the integral over this interval is zero, similarly between two and three. But for 𝑥 between three and nine, there is an area above the 𝑥-axis, which will cancel out the area below. In this case, with this value of 𝑥, there is more area above the 𝑥-axis than below it. And so the total integral is positive. But if we choose 𝑥 to be equal to four, then the areas exactly cancel. The integral between two and four of this function 𝑔 double prime is zero.
And hence, we see that four is a solution of the integral between two and 𝑥 of 𝑔 double prime of 𝑥 d𝑥 equals zero. Four is, therefore, the answer to our question. It’s the other value of 𝑥 on the interval from zero to 12 for which 𝑔 prime of 𝑥 equals zero. This is because, by the fundamental theorem of calculus, 𝑔 prime of four equals 𝑔 prime of two plus the integral between two and four of its derivative 𝑔 double prime of 𝑥 with respect to 𝑥.
We were told in the question that 𝑔 prime of two is zero. And we saw from the graph that the integral between two and four of 𝑔 double prime of 𝑥 with respect to 𝑥 was also zero. So 𝑔 prime of four really is zero.
Are there any other values of 𝑥 for which 𝑔 prime of 𝑥 equals zero? Well, we were given another solution in the question. 𝑔 prime of two is zero. And we can, by the way, see that this does make this integral zero. The interval has zero width as the upper limit is equal to the lower limit. And so there is zero area under the graph.
Are there any other values of 𝑥 which make this integral zero? Well, the question suggests not. We’re just asked for a single other value of 𝑥. But we can also see this from the graph. If we increased 𝑥 to some value below nine, then we’d just be adding area. And so our integral would be positive. If we increased 𝑥 beyond nine, we’d get some area below the 𝑥-axis. But even increasing 𝑥 all the way to 12, we don’t get enough area below the 𝑥-axis to cancel out all the extra area above it. The only other value of 𝑥 on the interval from zero to 12 for which 𝑔 prime of 𝑥 is equal to zero is therefore four.
